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. Author manuscript; available in PMC: 2020 Oct 9.
Published in final edited form as: J Geophys Res Planets. 2020 Apr 30;125(6):e2019JE006266. doi: 10.1029/2019je006266

FIGURE 9.

FIGURE 9.

Evolution of the rate of change of libration amplitude (Θ/ Θ) using the constant-Q tidal model for a synchronously rotating moon, with tidal expressions and associated tidal A constant defined by eqns. (12) and (21) to (40) in Wisdom and Tian (2015), for Lo = 2LEM, Q⊕ = 400, and varied A values. With a constant-Q model and A = 1.7 and 2, equation (6.9) predicts an extended period of libration amplitude damping (Θ/ Θ < 0) even as the Moon’s semi-major axis contracts (orbit contraction for these cases commences at t ≤ 15 [104 year]). This implies protracted resonance occupancy, consistent with simulations of Wisdom & Tian for this narrow range of A values (e.g., their Figures 2 and 9). In contrast, for A ≥ 3 (darker blue lines) increasingly strong amplitude excitation is predicted, suggesting limited resonance occupancy. Wisdom & Tian found minimal or no libration in formal evection for constant-Q tides and these larger A values.